Interactions of Classical and Numerical Algebraic Geometry

代数几何是一门经典学科，多年来一直处于代数、数论、几个复变量和几何的交叉点。个人计算机的出现，尤其是符号计算软件的发展，为这门学科引入了一个新的方面；突然间，人工无法处理的复杂计算成为可能。最近，在Sommese、Verschelde和Wampler的工作推动下，代数几何的数值方法被开创。这个新领域被恰当地称为数值代数几何。该领域的基本技术被称为同伦延拓，它提供了一种数值方法来跟踪由多项式方程组的代数变形到另一个多项式方程组所引起的几何变化。多项式系统在科学和工程中的普遍存在意味着数值代数基因组技术可以用于解决许多不同学科中出现的问题，例如运动学、化学、经济学、机器人视觉和电力电子学。近年来人们认识到，数值代数几何的技术也可以用于研究经典代数几何中的问题。这次会议的目的是将经典和数值代数几何学者聚集在一个活跃的讨论论坛上，以激发这两个社区之间的共同努力。这两个领域的交叉已经成熟，可以通过共同努力来快速发展，因此组织者希望这些互动将有助于在这方面取得重要进展。会议将于2008年5月22日至5月24日在圣母大学举行。在这三天里，将有13场讲座，由来自欧洲和美国不同地区的知名学者主讲。由于这一领域似乎即将产生许多新的研究途径，因此将积极寻求年轻数学家的参与。

Algebraic geometry is a classical discipline which has for many years been situated at the intersection of algebra, number theory, several complex variables, and geometry in all its incarnations. The advent of personal computing, and more so the development of software for symbolic computation, introduced a new facet of the discipline; it was suddenly possible to carry out computations far too sophisticated to be handled manually. More recently, a numerical approach to algebraic geometry was pioneered, largely driven by the work of Sommese, Verschelde, and Wampler. This new field is aptly known as numerical algebraic geometry. The fundamental technique of this field, known as homotopy continuation, provides a numerical means of tracking the changing geometry resulting from algebraically morphing one system of polynomial equations into another. The ubiquity of polynomial systems throughout the sciences and engineering means that the techniques of numerical algebraic geome try may be used to solve problems arising in many different disciplines, such as kinematics, chemistry, economics, robot vision, and power electronics.It has recently been realized that the techniques of numerical algebraic geometry may be used to study problems in classical algebraic geometry as well. The purpose of this conference is to bring together classical and numerical algebraic geometers in a lively discussion forum to spark joint efforts between these two communities. The intersection of these two fields is ripe for rapid growth fueled by joint work, so it is the hope of the organizers that these interactions will help to yield important advances along this front. The conference will take place at the University of Notre Dame from May 22 to May 24, 2008. There will be 13 lectures over those three days, given by eminent scholars from both communities and from various locations throughout Europe and the U.S. As this field seems poised to spawn many new avenues of research, the participation of young mathematicians will be actively sought.